The advent of VLSI digital IC technologies has made it attractive to perform many signal processing functions in the digital domain placing important emphasis on A/D and D/A conversions. Oversampling sigma-delta converters composed of simple and relatively high-tolerance analog components have recently become popular because they avoid a lot of difficulties encountered with conventional methods for A/D and D/A conversion. Classical sigma-delta modulators as shown in FIGS. 1(a), 1(b) and 1(c) use the technique of oversampling and noise shaping to move most of the quantization noise into high frequency band, well outside the band of the signal. Then, with a low-pass filter and decimator, we can easily filter out the high frequency noise such that the SNR at the signal band is significantly increased. In addition, the sigma-delta modulator (SDM) can trade resolution in time for that in amplitude such that imprecise analog circuits may be used. The use of high-frequency modulation and demodulation can eliminate the need for abrupt cutoffs in analog anti-aliasing prefilters at the input to the A/D converter, as well as in the smoothing postfilters at the analog output of the D/A converter. Besides, the performance of SDM is insensitive to nonideal effects, such as analog component matching or amplifier imperfection.
However, instability is a serious problem for high order SDM. The limitation of the high order SDM stems from the fact that high order integration cannot be realized due to the oscillation of the feedback loop. In this case, the modulator would settle into a large-amplitude low-frequency limit and result in instability. To improve the stability of higher-ordered SDM, three-stage MASH configuration is proposed in U.S. Pat. No. 4,704,600 (1987). MASH as shown in FIGS. 2(a) and 2(b) is a promising architecture to permit high-ordered noise shaping factor without instability problem, because it can offer an always stable modulation. MASH composes of several (first-ordered) SDM in cascade. The input of the next stage SDM is the quantization noise of the previous stage SDM. The quantization noises of the intermediate stage SDM are then all digitally cancelled. Thus only the quantization noise from the last stage SDM is left and MASH becomes always stable. However, there are still some defects in MASH architecture. For example, the quantization noise cancellation is sensitive to the gain matching accuracy between each stage of MASH. In addition, more operational amplifiers and more capacitors are required in MASH than in classical architecture such that the chip size of MASH increases.
In a classical second-ordered sigma-delta modulator as shown in FIG. 1(b), its transfer function is: EQU Y=X+(1-Z.sup.-1).sup.2 Q
Y: digital output X: analog input Q: quantization noise
In view of the above transfer function, the resolution of classical sigma-delta modulator is predominantly governed by order of noise shaping function and oversampling ratio.
Also, the transfer function of the two-stage MASH as shown in FIG. 2(b) is as follows: ##EQU1## In view of the above transfer function, the two-stage MASH offers second-ordered noise shaping factor as classical second-ordered SDM. But highly stable characteristic of MASH is identical to the first-ordered SDM.